A dichotomy characterizing analytic digraphs of uncountable Borel chromatic number in any dimension

TL;DR

This paper extends the Kechris-Solecki-Todorcevic dichotomy to higher dimensions, showing it holds for finite and infinite cases when using Baire-measurable homomorphisms, but not with continuous ones.

Contribution

It demonstrates the extension of the dichotomy to any dimension and clarifies the conditions under which it holds or fails.

Findings

Extension is possible in finite dimensions with continuous homomorphisms.

Extension fails in infinite dimensions with continuous homomorphisms.

The extension holds in infinite dimensions when using Baire-measurable homomorphisms.

Abstract

We study the extension of the Kechris-Solecki-Todorcevic dichotomy on analytic graphs to dimensions higher than 2. We prove that the extension is possible in any dimension, finite or infinite. The original proof works in the case of the finite dimension. We first prove that the natural extension does not work in the case of the infinite dimension, for the notion of continuous homomorphism used in the original theorem. Then we solve the problem in the case of the infinite dimension. Finally, we prove that the natural extension works in the case of the infinite dimension, but for the notion of Baire-measurable homomorphism.